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These average times to coalescence for two lineages in the context of two demes are both simple expressions that are easy to interpret. Equation 4.77 is a bit surprising since it says that the average time to coalescence for two lineages in the same deme is independent of the migration rate and is simply a function of the total population size as in a panmictic population (note that if each of d demes contains 2Ne lineages the total population size is NT = 2Ned). We can understand why this is the case by imagining what happens as the migration rate changes. If the migration rate decreases, the probability that a lineage migrates into another deme decreases with the effect of shortening the time to coalescence. However, in those cases when a migration event does occur, the lineage would take a longer time to migrate back before coalescence. The average time to coalescence is independent of the migration rate since these two factors exactly balance as the migration rate changes. When two lineages are in different demes the average time to coalescence increases as the migration rate decreases and as the number of demes increases. The average coalescence time is inversely proportional to the migration rate since migration is required to put two lineages into the same deme by chance. As the number of demes increases there are an increasing number of places for two lineages to be apart so that more migration events will have to occur until two lineages are together in the same deme.

Average coalescence times within demes and in the total population can also be used to express the degree of population structure. Earlier in the chapter we used probabilities of autozygosity to express population structure as a difference between the chance that two alleles sampled from the total population are different in state (HT) and the chance that two alleles sampled from a subpopulation are different in state Ht — HS

(T(0 1) = 2Ned). Putting these two average coalescence times together,